Elastic modulus vs Hardness plot

Another way to visualize the distribution of mechanical property results is to plot for example the elastic modulus (E) values vs the hardness (H) values. Such a plot leads sometimes to the observation of families of points and the definition of “sectors” or “bubbles”, each one corresponding to a single phase (e.g. soft matrix vs hard and stiff particles).

The correlation between elastic and plastic properties has been extensively studied in the literature [8], [3], [18] and [13].

Note

Elastic modulus is an intrinsic material property and hardness is an engineering property, which can be related to yield strength for some materials.

E-H map sectorization

As a first analysis of such a plot, sectors can be defined by giving an average value of elastic modulus and an average of hardness value, separating respectively by an horizontal line and a vertical line the different bubbles of points. Each sector is defined by a unique color.

Finally, average values of mechanical properties are given for each sectors directly into the graph, and a 4 color-coded map corresponding to this plot can be generated (see 2nd figure).

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Figure 32 Example of sectorized elastic modulus vs hardness plot

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Figure 33 Sectorized elastic modulus vs hardness plot with mean values and corresponding mechanical map

Automated cluster analysis (K-means, Gaussian Mixture, …)

Cluster analysis (or clustering) is an unsupervised machine learning task. For example, K-Means and Gaussian Mixtures (GMs) are both popular cluster analysis model for nanoindentation datasets.

K-Means models

The K-Means are often used for nanoindentation data clustering [12], [11], [1] and [10]. Like explains in [12], “the K-Means algorithm aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. The number of clusters is known and each point can belong only in one cluster. At first, random k cluster center points are generated and each data point is assigned to the cluster with the nearest center point (smallest Euclidean distance). Then, the mean of each cluster is calculated and the k cluster centers are replaced by the corresponding cluster mean. Again, each point is assigned to the nearest cluster, measured in Euclidean distance.” This method is as well described in the Matlab documentation [15].

A possible Matlab third party code, which could be used to define clusters with K-Means model is: https://www.mathworks.com/matlabcentral/fileexchange/24616-kmeans-clustering?s_tid=mwa_osa_a

Gaussian mixture models

The GMs model are as well used for nanoindentation data clustering [21] and [6]. This method is well described in the Matlab documentation [14], [16] and [17] but also in the literature [7].

This method is powerful to separate contribution of 2 or 3 phases (especially in the case of a soft metallic matrix with hard ceramic particles) in the cloud of experimental points [9]. Average mechanical property values can also be extracted using this method and a 2 or 3 color map can be obtained too.

The influence of indentation size and spacing on statistical phase analysis has also been studied by fast mapping indentation anc clustering analysis [4].

The Matlab third party code used to define clusters with GMs model is: GMMClustering.m

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Figure 34 Elastic modulus vs hardness plot with clusters of points obtained with GMM

Determination of the number of clusters

The optimal number of clusters (k) can be determined using the elbow method. This method looks at the total within-cluster sum of square (WSS) as a function of the number of clusters. One should choose a number of clusters so that adding another cluster doesn’t improve much better the total WSS.

This method can be defined as follow in 4 steps: 1) Compute clustering algorithm (e.g., K-Means or GMs algorithms) for different values of k. 2) For instance, by varying k from 1 to 5 clusters. For each k, calculate the total WSS. 3) Then, plot the curve of WSS according to the number of clusters k. 4) Finally, the location of a bend (knee) in the plot is generally considered as an indicator of the appropriate number of clusters.

Note that, the elbow method is sometimes ambiguous. Alternatives are for example the average silhouette method or the gap statistic method…

Next steps: Ashby map or self-organized maps

The next step after the different analytical, sectorization, clustering… approaches, could be to use nanoindentation outputs (phase mechanical properties in the case of a composite, an alloy…) for material selection, material design, material discovery….

A common strategy for material selection is the usage of conventional Ashby map [2]. An example of a typical Ashby map is given afterwards with materials families (envelops around different materials), using the CES Selector 2018 software [5]. At some point, it is possible to add material reference (bulk, homogeneous, monophasic, …) values on the E-H map, in order to compare experimental data with data from the literature.

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Figure 35 Typical Ashby map of elastic modulus vs Vickers hardness, obtained using CES Selector software

Regarding material design or material discovery, an option is the usage of self-organized maps (SOMs) [19] in the framework of material informatics. Here an example for Atomic Force Microscopy (AFM) technique [20].

References